Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral; application to the calculus of Δ-antiderivatives

This paper is devoted to the study of the Lebesgue Δ-integral on time scales. We give a formula in which such an integral is obtained as a sum of adequate real Lebesgue integrals. By using the relationship between these kinds of integrals and the Riemann ones, we rewrite the given expression to obtain the Δ-antiderivatives of functions defined on time scales. The results obtained are illustrated with some examples.